210 Trans. Acad. Sci. of St. Louis. 



From this and equation (15) 



/ m \ 



sin ( — , (7r — a) 



r sin (77 — <a) 



In the limit, when co = it, 



r A O' m 

 r' ~ A O' " m ' 

 or 



AG' X m = A'O'Xm ( 16 ). 



If m = m', equation ( 15) becomes 



a = (tt — co) (17). 



This is the equation of the perpendicular bisector to A A. 

 In this case (when m = m ) and 0' coincide and are at the 

 middle point of A A. 



If we assume that the rotating lines extend in both direc- 

 tions from their axes, and that w and &>' can have any values 

 from plus infinit} T to minus infinity, equation (1) or (3) rep- 

 resents a curve which may have a number of loops and 

 infinite branches, depending upon the relative values of m 

 and m , and equation (2) or (4) represents a curve which may 

 have, depending upon the relative values of m and m , a num- 

 ber of infinite branches, of which some pass through A and 

 the rest through A, but none of which pass through both A 

 and A. 



Consider first the case in which the lines rotate in the 

 same direction, which is represented by equation (3) or (1). 



1/1 1Tl ' 8 



In Fig. 5, in which _ = — ? = — numerically, AG, at angle 

 hi ni. 3 



a = 50' with AO, and A A, coinciding with AG, are the 



initial positions of the rotating lines AP aud AP. As 



these lines rotate in a counter clockwise direction, the part 



marked 1, and having A G as a tangent at A, is traced. This 



part approaches the asymptote marked /. After a position 



